# Experimental boundaries for size of electron?

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There is some confidence that electron is a perfect point e.g. to simplify QFT calculations. However, searching for experimental evidence (stack), Wikipedia article only points argument based on g-factor being close to 2: Dehmelt's 1988 paper extrapolating from proton and triton behavior that RMS (root mean square) radius for particles composed of 3 fermions should be ~g-2:

Using more than two points for fitting this parabola it wouldn't look so great, e.g. neutron (udd) has g~ -3.8, $$<r^2_n>\approx -0.1 fm^2$$

And while classically g-factor is said to be 1 for rotating object, it is for assuming equal mass and charge density. Generally we can classically get any g-factor by modifying charge-mass distribution:

$g=\frac{2m}{q} \frac{\mu}{L}=\frac{2m}{q} \frac{\int AdI}{\omega I}=\frac{2m}{q} \frac{\int \pi r^2 \rho_q(r)\frac{\omega}{2\pi} dr}{\omega I}= \frac{m}{q}\frac{\int \rho_q(r) r^2 dr}{\int \rho_m(r) r^2 dr}$

Another argument for point nature of electron is tiny cross-section, so let's look at it for electron-positron collisions:

Beside some bumps corresponding to resonances, we see a linear trend in this log-log plot: 1nb for 10GeVs (5GeV per lepton), 100nb for 1GeV. The 1GeV case means $$\gamma\approx1000$$, which is also in Lorentz contraction: geometrically means gamma times reduction of size, hence $$\gamma^2$$ times reduction of cross-section - exactly as in this line on log-log scale plot.

More proper explanation is that it is for collision - transforming to frame of reference where one particle rests, we get $$\gamma \to \approx \gamma^2$$. This asymptotic $$\sigma \propto 1/E^2$$ behavior in colliders is well known (e.g. (10) here) - wanting size of resting electron, we need to take it from GeVs to E=511keVs.

Extrapolating this line (no resonances) to resting electron ($$\gamma=1$$), we get 100mb, corresponding to ~2fm radius.

From the other side we know that two EM photons having 2 x 511keV energy can create electron-positron pair, hence energy conservation doesn't allow electric field of electron to exceed 511keV energy, what requires some its deformation in femtometer scale from $$E\propto1/r^2$$:

$\int_{1.4fm}^\infty \frac{1}{2} |E|^2 4\pi r^2 dr\approx 511keV$

Could anybody elaborate on concluding upper bound for electron radius from g-factor itself, or point different experimental boundary?

Does it forbid electron's parton structure: being "composed of three smaller fermions" as Dehmelt writes? Does it also forbid some deformation/regularization of electric field to a finite energy?

Edited by Duda Jarek

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4 hours ago, Duda Jarek said:

Another argument for point nature of electron is tiny cross-section, so let's look at it for electron-positron collisions:

Beside some bumps corresponding to resonances, we see a linear trend in this log-log plot: 1nb for 10GeVs (5GeV per lepton), 100nb for 1GeV. The 1GeV case means γ1000 , which is also in Lorentz contraction: geometrically means gamma times reduction of size, hence γ2 times reduction of cross-section - exactly as in this line on log-log scale plot.

Why would that reduce the physical cross-section? The length contraction is along the direction of travel.

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No matter interpretation, if we want boundaries for size of electron, it shouldn't be calculated for Lorentz contracted electron, but for resting electron - extrapolate above plot to gamma=1 ... or take direct values:

So what boundary for size of (resting) electron can you calculate from cross-sections of electron-positron scattering?

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12 minutes ago, Duda Jarek said:

No matter interpretation, if we want boundaries for size of electron, it shouldn't be calculated for Lorentz contracted electron, but for resting electron - extrapolate above plot to gamma=1 ... or take direct values:

So what boundary for size of (resting) electron can you calculate from cross-sections of electron-positron scattering?

In any collision, you can assume one of them is at rest.

Also, charged particle scattering will depend on the interaction time, so why wouldn't you expect a reduction in cross-section, as the energy went up, regardless of possible relativistic effects?

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So can you say something about electron size based on electron-positron cross section alone?

Edited by Duda Jarek

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1 hour ago, Duda Jarek said:

So can you say something about electron size based on electron-positron cross section alone?

Probably. A particle with some diameter will have a larger cross-section than a point particle, for a given interaction time. You can extrapolate to get the physical size of the electron from it, and AFAIK it's consistent with zero.

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Could you give some number? Article?

We can naively interpret cross-section as area of particle, but the question is: cross-section for which energy should we use for this purpose?

Naive extrapolation to resting electron (not Lorentz contracted) suggests ~2fm electron radius this way (which agrees with size of needed deformation of electric field not to exceed 511 keVs energy).

Could you propose some different extrapolation?

Edited by Duda Jarek

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1 hour ago, Duda Jarek said:

Could you give some number? Article?

We can naively interpret cross-section as area of particle, but the question is: cross-section for which energy should we use for this purpose?

You can always reference a cross section at some energy, and have a formula for energy dependence, if you know how it varies. You wouldn't necessarily reference it to zero, though; that would be a problem for a cross section that depended inversely on energy in some way.

1 hour ago, Duda Jarek said:

Naive extrapolation to resting electron (not Lorentz contracted) suggests ~2fm electron radius this way (which agrees with size of needed deformation of electric field not to exceed 511 keVs energy).

Could you propose some different extrapolation?

A 2 fm electron radius would likely have a noticeable correction needed for the QM solutions to atomic spectra, since the charge would be spread out accordingly.

Scattering cross section and physical size are not the same thing.

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Sure, so here is the original Cabbibo electro-positron collision 1961 paper: https://journals.aps.org/pr/abstract/10.1103/PhysRev.124.1577

Its formula (10) says sigma ~ \beta/E^2 ... which extrapolation to resting electron gives ~ 2fm radius.

Indeed it would be great to understand corrections to potential used in Schrodinger/Dirac, especially for r~0 situations like electron capture (by nucleus), internal conversion or positronium.

Standard potential V ~ 1/r goes to infinity there, to get finite electric field we need to deform it in femtometer scale

Edited by Duda Jarek
Latex problem

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1 hour ago, Duda Jarek said:

Sure, so here is the original Cabbibo electro-positron collision 1961 paper: https://journals.aps.org/pr/abstract/10.1103/PhysRev.124.1577

Its formula (10) says sigma ~ \beta/E^2 ... which extrapolation to resting electron gives ~ 2fm radius.

That's an interaction cross-section, not a size

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Sure, it isn't - fm size is only a suggestion, but a general conclusion here is that cross section does not offer a sub-femtometer boundary for electron size (?)

Dehmelt's argument of fitting parabola to 2 points: so that the third point is 0 for g=2 ... is "proof" of tiny electron radius by assuming the thesis ... and at most criticizes electron built of 3 smaller fermions.

So what experimental evidence bounding size of electron do we have?

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Arnold Neumaier has responded on stack ( https://physics.stackexchange.com/questions/397022/experimental-boundaries-for-size-of-electron ) - he has gathered many materials on this topic:
https://www.mat.univie.ac.at/~neum/physfaq/topics/pointlike.html
But still no clear argument that electron is much smaller then femtometer (?)

Anyway, to better specify the problem, define E(r) as energy in a radius r ball around electron.
We know that E(r) ~ 511keVs for large r, for smaller it reduces e.g. by energy of electric field. Assuming perfect point charge, we would get E(r) -> -infinity for r->0 this way. Where does divergence from this assumption starts?
More specifically: for example where is maximum of E'(r) - in which distance there is maximal deposition of 511keVs energy?
Or median range: such that E(r) = 511/2 keVs.
It is not a question about the exact values, only their scale: ~femtometer or much lower?

Edited by Duda Jarek

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2 hours ago, Duda Jarek said:

Arnold Neumaier has responded on stack ( https://physics.stackexchange.com/questions/397022/experimental-boundaries-for-size-of-electron ) - he has gathered many materials on this topic:
https://www.mat.univie.ac.at/~neum/physfaq/topics/pointlike.html
But still no clear argument that electron is much smaller then femtometer (?)

Anyway, to better specify the problem, define E(r) as energy in a radius r ball around electron.
We know that E(r) ~ 511keVs for large r, for smaller it reduces e.g. by energy of electric field. Assuming perfect point charge, we would get E(r) -> -infinity for r->0 this way. Where does divergence from this assumption starts?
More specifically: for example where is maximum of E'(r) - in which distance there is maximal deposition of 511keVs energy?
Or median range: such that E(r) = 511/2 keVs.
It is not a question about the exact values, only their scale: ~femtometer or much lower?

I think his argument is backwards. Why would you extrapolate to electrons at rest? The scattering is electrostatic, and the slower they go, the more time they spend near each other, interacting. It's when they go fast, and spend less time interacting, that you are reducing the effect of the electromagnetic interaction. If there were any "hard sphere" scattering, you would see a deviation from the trend. It's not there.

The argument about length contraction is suspect. Length contraction is in the direction of motion. It does not "shrink" the electron in all directions. It makes a sphere into a pancake, not a smaller sphere. (and that has ramifications for scattering, as well. For particles with structure, that has to factor into their calculations)

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Generally, we are interested in size of rest electron, not of squeezed electron.

There are some complex dependencies from its squeezing with Lorentz contraction - we need to remove them, e.g. by extrapolating to rest energy (any other ways?)

A general question regards distribution of electron's 511keVs energy - some of it is in energy of electric field (... infinite assuming perfect point), some could be e.g. in energy of fields related to other interactions electron takes part: gravitational, weak ...

So e.g. ball of which radius contains half of 511keVs energy of electron? Is it femtomer-scale radius, or much smaller?

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31 minutes ago, Duda Jarek said:

Generally, we are interested in size of rest electron,

There is no such thing as a rest electron.

See Earnshaw's theorem.

The 'size' of a quantum particle is totally dependent upon its environment, since its quantum expression extends over the entire quantum domain, perhaps to infinity.

That is how quantum tunnelling works for instance in this computer of mine.

Edited by studiot

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Ok, you can say that there are some quantum or statistical fluctuations ... I can respond with just adding "on average".

For example: ball of which radius contains on average half of 511keVs energy of electron? Is it femtomer-scale radius, or much smaller?

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7 minutes ago, Duda Jarek said:

Ok, you can say that there are some quantum or statistical fluctuations ... I can respond with just adding "on average".

For example: ball of which radius contains on average half of 511keVs energy of electron? Is it femtomer-scale radius, or much smaller?

I didn't mention the word fluctuations.

Was that post an answer to mine?

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1 hour ago, Duda Jarek said:

Generally, we are interested in size of rest electron, not of squeezed electron.

So why are you looking at the cross section of the electrostatic interaction, which does not tell you the size of the electron? scattering cross-section ≠ size

Quote

There are some complex dependencies from its squeezing with Lorentz contraction - we need to remove them, e.g. by extrapolating to rest energy (any other ways?)

How does that remove the effects of the electric field, which does not tell you the size of the electron?

Quote

A general question regards distribution of electron's 511keVs energy - some of it is in energy of electric field (... infinite assuming perfect point), some could be e.g. in energy of fields related to other interactions electron takes part: gravitational, weak ...

So e.g. ball of which radius contains half of 511keVs energy of electron? Is it femtomer-scale radius, or much smaller?

That sounds like the classical electron radius, or at least suffers from the same problem: trying to solve a quantum mechanical problem using classical physics.

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studiot, quantum formalism can be translated into probabilities with Born rule - while we cannot ask about exact e.g. position, QM still allows to ask about its expected value: "on average".

swansont, looking at electron-positon scattering cross section as one of suggestions, it includes all their interactions.

Electron's 511keV energy is at least partially distributed into energy of fields of interactions: probably mainly EM. Still we don't know this distribution (even average), naive assumption of perfect point would mean infinite energy.

So what is this configuration of EM field near the center of electron? E.g. in ball of what radius there is half of this energy? (on average)

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On re-reading this thread I see that we are just covering the same ground all over again.

Do you have a NEW argument, that does not rely on the scattering cross-section being conflated with the size?

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As written, I have returned to this topic due to Neumaier's page with many materials: https://www.mat.univie.ac.at/~neum/physfaq/topics/pointlike.html

But generally the fundamental question of size of electron remains unanswered - while there are many suggestions of femtometer-scale size of electron (as deformation from perfect point charge), I still haven't seen any real (not fitting parabola to two point) arguments that it is essentially smaller (claimed e.g. in Wikipedia).

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38 minutes ago, Duda Jarek said:

As written, I have returned to this topic due to Neumaier's page with many materials: https://www.mat.univie.ac.at/~neum/physfaq/topics/pointlike.html

But generally the fundamental question of size of electron remains unanswered - while there are many suggestions of femtometer-scale size of electron (as deformation from perfect point charge), I still haven't seen any real (not fitting parabola to two point) arguments that it is essentially smaller (claimed e.g. in Wikipedia).

48 minutes ago, Duda Jarek said:

studiot, quantum formalism can be translated into probabilities with Born rule - while we cannot ask about exact e.g. position, QM still allows to ask about its expected value: "on average".

Once again you avoided my comments.

Please give me a straightforward answer to a straightforward question, without second guessing my meaning, which is clearly not what you are guessing it to be.

I clearly stated that your statement about rest electrons is false due to Earnshaw's theorem.

I clearly stated that the wave function has a value at every point in the domain of interest, whthere it is interpreted as a probability or not.
Claiming fluctuations or random variations or probability does not alter this.

Or are you claiming that non tunneling electrons do not have a wavefunction that extends beyond their situs?

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56 minutes ago, Duda Jarek said:

swansont, looking at electron-positon scattering cross section as one of suggestions, it includes all their interactions.

Electron's 511keV energy is at least partially distributed into energy of fields of interactions: probably mainly EM. Still we don't know this distribution (even average), naive assumption of perfect point would mean infinite energy.

So what is this configuration of EM field near the center of electron? E.g. in ball of what radius there is half of this energy? (on average)

I mentioned the classical electron radius. That is one made using naive assumptions. Using classical physics to deduce quantum behavior is fraught with problems.

48 minutes ago, Duda Jarek said:

But generally the fundamental question of size of electron remains unanswered - while there are many suggestions of femtometer-scale size of electron (as deformation from perfect point charge),

The electric dipole measurements, if you look at them as a deformation from spherical charge, puts the limit at something like 10^-30 m.

48 minutes ago, Duda Jarek said:

I still haven't seen any real (not fitting parabola to two point) arguments that it is essentially smaller (claimed e.g. in Wikipedia).

I think that is a gross mischaracterization of the g-2 measurement. AFAICT there is no "parabola fit" as you have described. That's not how the result was obtained.

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studiot, according to Wikipedia, Earnshaw's theorem says "that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges" ... while here we are talking about configuration of EM field of a single electron.

Regarding " Or are you claiming that non tunneling electrons do not have a wavefunction that extends beyond their situs? " - no, by "on average" I mean that you can translate this wavefunction into probability to answer the question of size of electron, e.g. a radius such that half of 511keVs energy is on average in this radius around the center of electron.

swansont, if you want to relate radius of electron to its electric dipole moment ... why not to use magnetic dipole instead? - which is huge.

If you are able to defend Dehmelt's g-factor argument I would be really interested. It looks like at first he extrapolated with line, getting negative radius for electron - so he has chosen parabola to get exactly 0 radius for g=2, what is "proving" by assuming the thesis. Also, we have more 3-parton particles, like neutron, which RMS radius is negative due to minuses being further than pluses ...

For fundamental particles we cannot talk about RMS radius, but we can about differences from (infinite energy) EM field configuration of perfect point charge.